University of Florida/Egm6341/s11.TEAM1.WILKS/Mtg3

EGM6321 - Principles of Engineering Analysis 1, Fall 2010
Mtg 3: Thur, 26 Aug 10

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NOTE: - page numbering 3-1 defined as meeting number 3, page 1 - T = torque [[media: 2010_08_24_15_22_30.djvu | Fig.p.1-1]] - HW* [[media: 2010_08_26_13_57_13.djvu | Eq.(3)P.2-1]] : "Ordinary" Differential Equation (ODE) order = highest order of derivative Nonlinearity = What is linearity? ; use intuition for now, formal definition soon. System has 3 unknowns: $$ y^1(t) \ $$ $$ u^1(s,t) \ $$ $$ u^2(s,t) \ $$ $$ \equiv \ $$ Partial Differential Equations (PDE) 3 equations are coupled $$ \Rightarrow \ \ $$ Numerical Methods Simplify for analytical solution [[media: http://clesm.mae.ufl.edu/~vql/pdf/jam.1989.vuquoc.olsson.pdf| Ref:VQ&O 1989]] 2nd Order $$ \rightarrow \ \ $$ 2nd Order nonlinear $$ \rightarrow \ \ $$ linear unknown varying coefficient $$ \rightarrow \ \ $$ known varying coefficient

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Note: Math structure of coefficient $$ c_i(Y',t) \ $$ for $$ i=0,1,...3 \ $$ is known, but not their values until $$ u^1 \ $$ and $$ u^2 \ $$ are known (solved for) General structure of Linear 2nd order ODEs with varying coefficients (L2_ODE_VC) where $$ y''=\frac{d^2y}{dx^2} \ $$ $$ x= \ $$ independant variable $$ y(x)= \ $$ dependant variable (unknown function to solve for) Many applications in engineering are a result of solving PDEs by separation of variables. Some examples include, but are not limited to: Heat, Solids, Fluids, Acoustics and electrmagnetics. Examples of these types equations are: the Helmholz equation: $$ \Delta\ X+k^2X=0 \ $$ and the Laplace Equation: $$ \Delta\ X=0 \ $$ [[media: Egm6321.f09.mtg28.djvu| Ref F09 Mtg.28]], [[media: Egm6321.f09.mtg29.djvu| Ref F09 Mtg.29]] , [[media: Egm6321.f09.mtg30.djvu| Ref F09 Mtg.30]]

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In 3_D, $$ x=(x_1,x_2,x_3) \ $$ Where the lowercase $$ x \ $$ in the first term $$ X(x) \ $$ is defined as $$ x=(x_1,x_2,x_3) \ $$ and $$ X_1(x_1)X_2(x_2)X_3(x_3) \ $$ is the separation of variables Where $$ \xi\ \ $$ in the first term $$ X( \xi\ ) \ $$ is defined as $$ \xi\ =( \xi\ _1, \xi\ _2, \xi\ _3) \ $$ and $$ X_1( \xi\ _1)X_2( \xi\ _2)X_3( \xi\ _3) \ $$ is the separation of variables Separated equations for $$ i=1,2,3 \ $$

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Simplify: $$ \xi\ _i \rightarrow \ x \ $$ $$ X_i ( \xi\ _i) \rightarrow \ y(x) \ $$ $$ g_i ( \xi\ _i) \rightarrow \ g(x) \ $$ $$ f_i ( \xi\ _i) \rightarrow \ a_0(x) \ $$ [[media: 2010_08_26_14_53_13.djvu | Eq.(3)p.3-3]]: Where $$ \frac{g'(x)}{g(x)} = a_1(x) \ $$ Particular case of [[media: 2010_08_26_14_53_13.djvu | Eq.(1)p.3-2]] Linearity: Let $$ F(.) \ $$ be an operator. $$ u \ $$ and $$ v \ $$ are 2 possible arguments (could be functions) of $$ F(.) \ $$ $$ F( \alpha\ u+ \beta\ v = \alpha\ F(u) + \beta\ F(v) \ $$ Where $$ \alpha\ \ $$ and $$ \beta\ \ $$ are any arbitrary number.